Introduction to transmission dynamic models of infectious diseases

Ted Cohen (theodore.cohen@yale.edu) Department of Epidemiology of Infectious Diseases

Models – what are they? A model is a simplified description of a complex

entity or process

Why might we want a model?

•  To improve our fundamental understanding of how a system works

•  To predict or project

how the system will change over time (and possibly in response to manipulation)

Desirable properties of models

Simplicity Realism (understanding) (prediction)

Why do we use models for infectious diseases?

•  The dynamics of infectious diseases are complex

• Non-linearities – Small changes in input can produce large

changes in output

•  Emergent properties – More is different

Overview

Simplicity Realism (understanding) (prediction)

1. Compartmental Models

2. Other model types:

Metapopulation Individual-based

Spatial Network

1. Compartmental models

•  Population divided into categories defined by health/disease status

•  SIR model as a prototype

SIR model (no demography)

dS SdtdI S IdtdR Id

I

t

I

γ

β

β

γ

= −

= −

=

Closed system: no births/deaths S, I, R are proportions: S+I+R=1 Disease is approximated by SIR: no latency, immunity is complete

transmission parameter (transmission probability/contact*contact rate)

recovery rate (1/disease duration)

β

γ

=

=

R0: Basic reproductive number

•  Expected number of secondary cases of disease produced directly by an average infectious individual entering an entirely susceptible population

0

0

11

RR>

<

Epidemic occurs No epidemic occurs

Threshold phenomenon

•  Kermack & McKendrick 1927 •  Minimum fraction of population that is

susceptible necessary for an epidemic to occur.

( ) 0 (if epidemic occurs, 0)

0

( is the relative removal rate)

dI IS Idt

dII Sdt

S

S

β γ

β γ

β γ

γβ βγ

>

= −

− > >

− >

Threshold phenomenon & R0

0

1>

(Basic reproductive number defined when S=1)

>1

R

S

β

γβ

β

γ

γβ

γ

=

>

Threshold phenomenon & critical proportion to vaccinate

0

0

(threshold proportion of susceptibles necessary for an epidemic)

1

(there 11-fore, we must vaccinate at least of the population to prevent an epidemic)R

S

SR

γβ

>

>

Epidemic curve (SIR, no demography)

0

0

1/5

1/3

2/5

1/2

3/5

5/7

4/5

8/9

1

1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401

time

Proportion I:1

S:1R:1

( )dS S I R S SdtdI S I IdtdR I Rdt

I

I

µ

β

µ

γ

β

µ

µγ

= + + − −

= − −

= − transmission parameter (transmission probability/contact*contact rate)

recovery rate (1/disease duration)

mortality/fertility (1/life expectancy)

β

γ

µ

=

=

=

SIR model (with demography)

Fixed population size (deaths=births) No disease induced mortality

R0 in an SIR model with demography

0

(When is I increasing?: When the above equation is +.)( ) 0

1

dI IS I Idt

I S

R

β γ µ

βγ µ

β γ µ

β γ µ

βγ µ

= − −

−

+

>+

+

>

=

− >

Epidemic curve (SIR, with demography)

0

0

1/5

1/3

2/5

1/2

3/5

5/7

4/5

8/9

1

1 201 401 601 801 1001 1201 1401 1601 1801 2001 2201 2401

time

Proportion I:1

S:1R:1

Equilibrium condition: S

0

(At equilibrium the above equation = 0.)*( * ) 0

(Let's examine the non-trivial case when I* 0) *

*

1*

dI IS I Idt

I S

S

R

S

S

β γ µ

β γ µ

β γ µ

γ µβ

= − −

− − =

≠

=

=

+

+=

Equilibrium condition: I

0

0

at equilibrium:

* * * 0

1substituting for S

( 1)*

* =

dS IS Sdt

I S S

RRI

µ β

µβ

µ

µ β µ

= − −

− − =

−=

Average age of infection (at equilibrium)?

( )

(Ignoring small rates of mortality/fertility)

rate of leaving S = mean duration of staying in S at equilibium = Average age of infection (A)

1 *

substituting for I* =

dS S I R IS Sdt

dS ISdt

I

AI

µ β µ

β

β

β

= + + − −

≈ −

≈

0

0

0

0

( -1) :

1 (where L = average lifespan)( -1) -1)

( -1

(

)

R

L RA

LAR R

µβ

µ

=

≈ =

R0 and the average age of infection

Type II mortality: constant hazard, exponentially distributed lifespansLA= (R0 -1)

Type I mortality: uniformly distributed lifespansLA= R0

Infectious diseases with higher R0 have lower average ages of infection

Rubella: a case study •  Pathogen: rubella virus (RNA virus); infection confers

immunity •  Transmission:

–  respiratory (aerosol) •  Clincal symptoms:

–  generally mild (fever, rash, arthralgia/arthritis) –  rarely more serious complications (encephalitis, hemorrhagic

manifestations)

•  Vertical transmission Congenital Rubella Syndrome (CRS) –  Infection of mother earlier in first trimester >85% babies will be

affected •  All organ systems involved (deafness, eye abnormalities, CV and

neurological defects) –  Indicates that fraction of women of child-bearing age who are

susceptible is very important

Anderson and May, J Hyg 1983

Gambia, 1966-1976 USA, 1966-1968 frac

tion

ever

infe

cted

frac

tion

ever

infe

cted

age age

Rubella intervention

•  Vaccination –  Goal is to reduce cases of CRS

•  Expected to have both individual-level and population-level benefits –  Individual-level: protect those who have been

vaccinated –  Population-level: indirectly protect those who have

not been vaccinated

•  Critical proportion of the population to vaccinate: –  1-(1/R0)

Possible perverse effects of a vaccination program?

•  What if we do not achieve herd immunity? –  Transmission persists (but lower force of infection)

–  Effect on the average age at infection? • Effect on expected number of CRS cases?

•  Effects explored in compartmental models by Knox (1980), Anderson and May (1983)

Cases of rubella after vaccination Cases of rubella before vaccination

Fraction immunized at birth

Among women 16-40 years old

Anderson and May, J Hyg 1983

Greece

•  MMR vaccine introduced for 1 year old (girls and boys) ~1975

•  No formal policies for achieving high coverage

–  Rubella vaccination classified as “optional” by MOH –  Rubella vaccination given only on request to girls

10-14 yo in public sector –  1980s: vaccine coverage for rubella consistently

below 50%

Panagiotopoulos BMJ 1999

Panagiotopoulos BMJ 1999

SIR models: What assumptions?

•  SIR states effectively summarize categories of people

•  Usually deterministic (but can include stochasticity)

•  Homogenous mixing •  Exponentially distributed

waiting times

•  In summary, heterogeneity is largely ignored

Can we include more “realism”? • Within compartmental model approach:

– Represent different natural history (SIS, SI, SEIR) – Demographic characteristics (age, sex) – Behavioral categories (high/low activity groups)

•  But, the number of compartments increases quickly – SEIR with 5 age groups, sex, and 2 activity

groups • 4*5*2*2=80 compartments!

2. We can increase complexity with other modeling approaches

S

R

I

S

I

R

Compartmental

Individual-based

Network

•  Stochasticity easily included and heterogeneity more naturally expressed in these types of models

•  Focus is on

experience of individuals, rather than on “classes” of individuals.

S

I

R

Questions/situations that often warrant other approaches*

•  Spatial spread of disease •  Explicit and detailed contact patterns between

individuals •  Modeling complex interventions (e.g. targeting

individuals via contact tracing) •  Elimination of pathogens •  Emergence of new pathogens in populations •  Nosocomial transmission •  Plus many others

*many of these issues can also be approximated using variations of compartmental models

Keeling J Roy Soc Interface 2005

Random Lattice Small-world

Spatial Scale-free

N=100 Average degree = 4

Keeling J Roy Soc Interface 2005

Random Lattice Small-world

Spatial Scale-free

N=10,000 100 epidemics; black is mean

100 shortcuts

10 shortcuts

20 shortcuts

For policy-making, more complex models are sometimes required

•  “Predictive” models • Comparing the performance of alternative “realistic” interventions

• Coupling with economic considerations to generate cost-effectiveness comparisons of different control strategies

Projection for pandemic influenza

Individual-based model: UK and US populations; international travel (seeding); air travel within US; transmission within households, schools, workplaces, and in the community

Ferguson et al, Nature 2006

Summary

•  Models of infectious diseases may be of various forms

•  The structure and approach should be dictated by the research question and availability of data

•  Both simple and more complex models have proven to be useful tools for understanding disease dynamics, projecting disease trends, and informing control policy